1.08d Evaluate definite integrals: between limits

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\), $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of \(I _ { 4 }\). By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$

10 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), show that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta - 2 \cos 4 \theta - \cos 2 \theta + 2 ) .$$ Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta\).
[0pt] [Question 11 is printed on the next page.]

11 Show that \(\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c\), where \(c\) is a constant. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$ Use the substitution \(x = \sin u\) to show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$ Find \(I _ { 4 }\).

4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.

1. Find the value of \(k\).
2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

6 Alex is investigating the area, \(A\), under the graph of \(y = x ^ { 2 }\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\). \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}

1. Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively.
2. Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.

   Width \(\delta x\)	0.1	0.05	0.025	0.0125
   Lower bound for area \(A\)	0.73	0.761	0.776	0.784
   Upper bound for area \(A\)	0.855	0.823	0.807	0.799

   Use Alex's results to estimate the value of \(A\) correct to \(\mathbf { 2 }\) significant figures. Give a brief justification for your estimate.
3. Write down an expression, in terms of \(y\) and \(\delta x\), for the exact value of the area \(A\).

Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)

1. show that \(3 k + 5 \sqrt { k } - 12 = 0\)
2. Hence, using algebra, find any values of \(k\) such that $$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$

1. Find the value of the constant \(k , 0 < k < 9\), such that

$$\int _ { k } ^ { 9 } \frac { 6 } { \sqrt { x } } \mathrm {~d} x = 20$$

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant. Given that the area of \(R\) is 10

1. find, in simplest form, the value of
   1. \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
   2. \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
2. show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.

7. Given that \(k \in \mathbb { Z } ^ { + }\) a. show that \(\int _ { 2 k } ^ { 3 k } \frac { 6 } { ( 7 k - 2 x ) } \mathrm { d } x\) is independent of \(k\),
b. show that \(\int _ { k } ^ { 2 k } \frac { 2 } { 3 ( 2 x - k ) ^ { 2 } } \mathrm {~d} x\) is inversely proportional to \(k\).

4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that $$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$ where \(k\) is a constant to be found.

16. \begin{figure}[h] \end{figure} Figure 8 shows a sketch of the curve with parametric equations $$x = 4 \cos t \quad y = 2 \sin 2 t \quad 0 \leq t \leq \frac { \pi } { 2 }$$ where \(t\) is a parameter.
The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 7.
a. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 16 \sin ^ { 2 } t \cos t \mathrm {~d} t$$ b. Hence, using algebraic integration, find the exact area of \(R\), giving in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.

5. a. Given that $$\frac { x ^ { 2 } - 1 } { x + 3 } \equiv x + P + \frac { Q } { x + 3 }$$ find the value of the constant \(P\) and show that \(Q = 8\) \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 } \quad x > - 3$$ Figure 3 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 5\).
b. Find the exact area of \(R\), writing your answer in the form \(a \ln 2\), where \(a\) is constant to be found.
(4)

Given that \(k \in \mathbb { Z } ^ { + }\)

1. show that \(\int _ { k } ^ { 3 k } \frac { 2 } { ( 3 x - k ) } \mathrm { d } x\) is independent of \(k\),
2. show that \(\int _ { k } ^ { 2 k } \frac { 2 } { ( 2 x - k ) ^ { 2 } } \mathrm {~d} x\) is inversely proportional to \(k\).

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \sin ^ { 2 } t \quad y = 2 \sin 2 t + 3 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 6, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 4\)

1. Show that the area of \(R\) is given by $$\int _ { 0 } ^ { a } \left( 8 - 8 \cos 4 t + 48 \sin ^ { 2 } t \cos t \right) \mathrm { d } t$$ where \(a\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { x } { 1 + \sqrt { } x }\)

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
2. Explain how the trapezium rule can be used to give a better approximation for the area of \(R\).
3. Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
   1. \(\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x\)
   2. \(\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x\)

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. Given that \(A\) is constant and

$$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 2 A ^ { 2 }$$ show that there are exactly two possible values for \(A\).

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 3 x\). The curve passes through the point (6, 20).

1. Determine the equation of the curve.
2. Hence determine \(\int _ { 1 } ^ { p } y \mathrm {~d} x\) in terms of the constant \(p\).

10 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.

1. Determine the values of \(p , q\) and \(r\).
2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) where \(v = ( 2 t + 1 ) ( 3 - t )\).

1. Find the deceleration of \(P\) when \(t = 4\).
2. State the positive value of \(t\) for which \(P\) is instantaneously at rest.
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = 4\).

12 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-09_647_935_260_242} A particle \(P\) moves in a straight line. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = - 3 t ^ { 2 } + 24 t + k\), where \(k\) is a positive constant. The diagram shows the velocity-time graph for the motion of \(P\). \(P\) attains its maximum velocity at time \(T\) seconds. Given that the distance travelled by \(P\) between times \(t = 1\) and \(t = T\) is 297 m , determine the time when \(P\) is instantaneously at rest. \section*{END OF QUESTION PAPER}